Centralizers and the maximum size of the pairwise noncommuting elements in nite groups

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1 Hacettepe Journal of Mathematics and Statistics Volume 46 (2) (2017), Centralizers and the maximum size of the pairwise noncommuting elements in nite groups Seyyed Majid Jafarian Amiri and Hojjat Rostami Abstract In this article, we determine the structure of all nonabelian groups such that has the minimum number of the element centralizers among nonabelian groups of the same order. As an application of this result, we obtain the sharp lower bound for ω() in terms of the order of where ω() is the maximum size of a set of the pairwise noncommuting elements of. eywords: nite group, centralizer, CA-group AMS Classication: 20D60 Received : Accepted : Doi : /HJMS Introduction and main results Throughout this paper will be a nite group and Z() will be its center. For a positive integer n, let Z n and D 2n be the cyclic group of order n and the dihedral group of order 2n, respectively. For a group, we dene Cent() = {C (x) : x } where C (x) is the centralizer of the element x in. It is clear that is abelian if and only if Cent() = 1. Also it is easy to see that there is no group with Cent() = 2 or 3. Starting with Belcastro and Sherman [7], many authors have investigated the inuence of Cent() on the group (see [1], [4], [6], [7], [17-21] and [27-29]). In the present paper, we describe the structures of all groups having minimum number of centralizers among all nonabelian groups of the same order, that is: Department of Mathematics, Faculty of Sciences, University of Zanjan, P.O.Box , Zanjan, Iran sm jafarian@znu.ac.ir Department of Mathematics, Faculty of Sciences, University of Zanjan, P.O.Box , Zanjan, Iran h.rostami5991@gmail.com

2 Theorem. Let be a nonabelian group of order n. If Cent() Cent(H) for all nonabelian groups H of order n, then one of the following holds: (1) is nilpotent, Cent() = p + 2 and Z() = Z p Z p where p is the smallest prime such that p 3 divides n. (2) is nonnilpotent, Cent() = p m + 2 and Z() = (Z p) m Z l where l > 0 and p m is the smallest prime-power divisor of n such that p m 1 and n are not relatively prime. The following corollary are immediate consequence of Theorem Corollary. Suppose that n is even and is a nonabelian group of order n. If Cent() Cent(H) for all nonabelian groups H of order n, then Cent() = 4 or p + 2 where p is the smallest odd prime divisor of n and also is isomorphic to one Z() of the following groups: Z 2 Z 2, Z p Z p, D 2p Remark. We notice that both conditions (1) and (2) of Theorem 1.1 may happen for some positive integer n. For example there exist two groups 1 and 2 of order 54 such that Cent( 1) = Cent( 2) = 5, 1 Z( 1 ) = Z 3 Z 3 and 2 Z( 2 ) = D 6. There are interesting relations between centralizers and pairwise noncommuting elements in groups (see Proposition 2.5 and Lemma 2.6 of [1]). Let be a nite nonabelian group and let X be a subset of pairwise noncommuting elements of such that X Y for any other set of pairwise noncommuting elements Y in. Then the subset X is said to have the maximum size, and this size is denoted by ω(). Also ω() is the maximum clique size in the noncommuting graph of a nite group. The noncommuting graph of a group is dened as a graph whose \ Z() is the set of vertices and two vertices are joined if and only if they do not commute. By a famous result of Neumann [22] answering a question of Erd s, the niteness of ω() is equivalent to the niteness of the factor group which follows that Cent() is nite. Also,if has a nite number of Z() centralisers, then it is easy to see that ω() is nite. Various attempts have been made to nd ω() for some groups. Pyber [24] has proved that there exists a constant c such that Z() cω(). Chin [13] has obtained upper and lower bounds of ω() for extra-special groups of odd order. Isaacs has shown that ω() = 2m + 1 for any extraspecial group of order 2 2m+1 (see page 40 of [11]). Brown in [9] and [10] has investigated ω(s n) where S n is the symmetric group on n letters. Also Bertram, Ballester-Bolinches and Cossey gave lower bounds for the maximum size of non-commuting sets for certain solvable groups([5]). Recently authors [17, 20] have determined all groups with ω() = 5 and obtained ω() for certain groups. nown upper bounds for this invariant were recently used to prove an important result in modular represention theory ( [13]). In this article we determine the structure of nonabelian groups of order n such that ω() ω(h) for all nonabelian groups H of order n Theorem. Let be a nonabelian group of order n. If ω() ω(h) for all nonabelian groups H of order n, then one of the following holds: (1) is nilpotent, ω() = p + 1 and Z() = Z p Z p where p is the smallest prime such that p 3 divides n. (2) is nonnilpotent, ω() = p m + 1 and Z() = (Z p) m Z l where l > 0 and p m is the smallest prime-power divisor of n such that p m 1 and n are not relatively prime. Throughout this paper we will use usual notation which can be found in [25] and [15].

3 Proofs of the main results The following lemmas are useful in the proof of the main theorem Lemma. Let, 1,, n be nite groups. Then 1. If H, then Cent(H) Cent() ; 2. If = n i, then i=1 Cent() = n Cent(i). i=1 Proof. The proof is clear. In Lemma 2.7 of [4], it was shown that if p is a prime, then Cent() p + 2 for all nonabelian p-groups and the equality holds if and only if Z() = Z p Z p. In the following we generalize this result for all nilpotent groups Lemma. Let be a nilpotent group and p be a prime divisor of such that a Sylow p-subgroup of is nonabelian. Then Cent() p + 2 with equality if and only if Z() = Z p Z p. Proof. Suppose that P is a Sylow p-subgroup of. Then we have Cent() Cent(P ) p + 2 by Lemma 2.1(1) and Lemma 2.7 of [4], as wanted. Now, assume that Cent() = p + 2. Since is nilpotent, each Sylow q-subgroup of is abelian for each prime divisor q p of by Lemma 2.1(2). Consequently Z() = P which is isomorphic to Zp Zp by Lemma 2.7 of [4]. The converse holds Z(P ) similarly. Recall that a minimal nonnilpotent group is a nonnilpotent group whose proper subgroups are all nilpotent. In 1924, O. Schmidt [26] studied such groups. The following result plays an important role in the proof of Theorem Lemma. Let be a minimal nonnilpotent group. Then Z() is Frobenius such that the Frobenius kernel is elementary abelian and the Frobenius complement is of prime order. Proof. By Theorem of [25], we have = P Q where P is a unique Sylow p-subgroup of and Q is a cyclic Sylow q-subgroup of for some distinct primes p and q. Also by Exercise of [25], the Frattini subgroups of P and Q are contained in Z(). It follows that P Z() is elementary and QZ() is of order q. Since all Sylow subgroups Z() Z() of are abelian, Theorem of [25] gives that ( Z() Z() ) Z( ) = 1. Since Z() P = [P, Q], we have P Z() Z() and so Z( ) is a q-group. On the other hand Z() Z() Z() since is not nilpotent, Z( ) = 1. Now it is easy to see that is a Frobenius Z() Z() group Proposition. Let = H be a Frobenius group such that H is abelian. Z() Z() Z() If Z() < Z(), then Cent() = Cent() and if Z() = Z(), then Z() Cent() = Cent() +. Also ω() = ω() + Z() Proof. See Proposition 3.1 of [18] and its proof. Z(). Recall that a group is a CA-group if the centralizer of every noncentral element of is abelian. R. Schmidt [26] determined all CA-groups (see Theorem A of [14]). Now we are ready to prove the main result. Proof of Theorem 1.1. Suppose that is a nilpotent group. Since is not abelian, a Sylow q-subgroup of is not abelian for some prime q. It follows from Lemma 2.2 that Cent() q + 2.

4 196 But there exists a nonabelian group H := Q Z n of order n where Q is a nonabelian q 3 group of order q 3 and we see that Cent(H) = q + 2. Since has the minimum number of the element centralizer, we must have Cent() = p + 2 and p must be the smallest prime such that p 3 divides n. Also Z() = Z p Z p by Lemma 2.2, as wanted. Now, assume that is a nonnilpotent group of order n. Then there exist two prime divisors q and r of n such that q divides r k 1 for some positive integer k by Corollary 1 of [23]. We claim that if p m is the smallest prime-power divisor of n such that gcd(p m 1, n) 1, then Cent() p m + 2. Since is nite and nonnilpotent, contains a minimal nonnilpotent subgroup M. It M follows from Lemma 2.3 that is Frobenius with the kernel and the complement Z(M) Z(M) H. Note that = Z(M) Z(M) pt H 1 and = p2 for some primes p1 and p2 such that Z(M) p 2 p t 1 1. It follows from Proposition 2.4 that Cent(M) +2 = Z(M) pt Since M is a subgroup of, we have Cent() Cent(M) p t which is equal or greater than p m + 2 by hypothesis. This proves the claim. Now we want to nd the structure of nonnilpotent groups for which the equality occurs. Assume that Cent() = p m + 2. We shall prove that Z() = (Z p) m Z l for some positive integer l. By hypothesis and the previous paragraph, there is a subgroup M M of such that is Frobenius with the kernel of order Z(M) Z(M) pm and the Frobenius H complement which is cyclic of prime order. Since + Cent() Cent(M) Z(M) Z(M) by Proposition 2.4 and Cent(M) Cent() = p m + 2, we have Cent(M) = p m + 2 and Cent() = 1. It follows that is abelian and so M is a CA-group by Theorem A (II) of [14]. Next, we show that is a CA-group. Since M is a CA-group, we have p m + 1 is the maximum size of a set of pairwise non-commuting elements of M by Lemma 2.6 of [1] and so the maximum size of a set of pairwise non-commuting elements of is at least p m + 1. Since Cent() = p m + 2, the maximum size of a set of pairwise non-commuting elements of must be p m + 1. Therefore is CA-group by Lemma 2.6 of [1]. Now we apply Theorem A of [14]. Note, rst, that if 8 divides = n, then by hypothesis Cent() Cent(D 8 Z n 8 ) = 4 and so Cent() = 4. Therefore Z() = Z 2 Z 2 by Fact 3 of [7] and so is nilpotent, a contradiction. Hence 8 does not divide n and so Cent() 5 by Fact 4 of [7]. Also if 6 divides n, then Cent() Cent(D 6 Z n 6 ) = 5 which implies Cent() = 5. Therefore Z() = D 6 by Fact [7] and so we have the result. Thus we may assume that 6 does not divide n. Now since is not nilpotent, satises (I), (II) or (III) of Theorem A of [14]. Therefore has an abelian subgroup A of prime index r or = T is a Frobenius group with the Frobenius kernel and the Z() Z() Z() Z() T Frobenius complement. In the rst case, we have Z() = p m by Theorem 2.3 of [6] and so = Z() pm r by Lemma 4 (page 303) of [8]. Consequently = A L Z() Z() Z() L where = r and A = Z() Z() pm. By the property of p m, the number of Sylow r- subgroup of is Z() pm and so is Frobenius. Again by the property of Z() pm A, is Z() characteristically simple which implies that it is elementary, as wanted. In the second case, it follows from (II)-(III) of Theorem A of [14] that T is abelian and is abelian or = QZ() where Q is a normal Sylow q-subgroup of for some prime q. If is abelian, then Z() < Z() and so Cent() = + 2 by Proposition 2.4. Z() Therefore = Z() pm. By the property of p m, is elementary. On the other hand Z() T is cyclic by Corollary 6.17 of [15] and so we have the result. Z() If = QZ(), then = Z() qa T and since is Frobenius, we have divides Z() Z() q a 1. It follows that p m q a by hypothesis. On the other hand p m + 2 = Cent()

5 by Proposition 2.4 and this implies that Z() pm = q a. It follows from Proposition 2.4 that Cent() = 1 and is abelian. The rest of the proof is similar to the previous case. Proof of Theorem 1.4 The proof is similar to the previous theorem. If is nilpotent, then ω() p + 1 where p is the smallest prime such that p 3 divides and the equality holds if and only if Z() = Z p Z p. Now suppose that is nonnilpotent. Then contains a minimal nonnilpotent M subgroup M and so = H is Frobenius such that = Z(M) Z(M) Z(M) Z(M) pm 1 and H = Z(M) p2 by Lemma 2.3. Since H is abelian and has at least pm 1 conjugates in, say H = H 1, H 2,, H p m 1, we see {x 1,, x p m 1 } is a subset of pairwise noncommuting elements of M where x i H i \ {1}. It follows that ω() p m p m + 1. The remainder of the proof is similar to Theorem 1.1. Acknowledgment. The authors would like to thank the referee for his/her careful reading and valuable comments. References [1] A. Abdollahi, S. M. Jafarian. Amiri and A. M. Hassanabadi, roups with specic number of centralizers, Houston J. Math. 33(1) (2007), [2] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algbera 298 (2) (2006), [3] A. Abdollahi, A. Azad, A. Mohamadi Hasanabadi and M. Zarrin, On the clique numbers of non-commuting graphs of certain groups, Algebra Colloq, 17(4) (2010), [4] A. R. Ashra, On nite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), [5] A. Ballester-Bolinches and J. Cossey, On non-commuting sets in nite soluble CC-groups, Publ. Mat. 56 (2012), [6] S. J. Baishya, On nite groups with specic number of centralizers, International Electronic Journal of Algebra, 13(2013), [7] S. M. Belcastro and. J. Sherman, Counting centralizers in nite groups, Math. Mag. 5 (1994), [8] Y.. Berkovich and E. M. Zhmu'd, Characters of Finite roups, Part 1, Transl. Math. Monographs 172, Amer. Math. Soc., Providence. RI, [9] R. Brown, Minimal covers of S n by abelian subgroups and maximal subsets of pairwise noncommuting elements, J. Combin. Theory Ser. A 49 (1988), [10] R. Brown, Minimal covers of S n by abelian subgroups and maximal subsets of pairwise noncommuting elements, II, J. Combin. Theory Ser. A 56 (1991), [11] E. A. Bertram, Some applications of graph theory to nite groups, Discrete Math. 44(1) (1983), [12] A. M. Y. Chin, On noncommuting sets in an extraspecial p-group, J. roup Theory 8(2) (2005), [13] A. Y. M. Chin, On non-commuting sets in an extraspecial p-group, J. roup Theory, 8.2 (2005), [14] S. Dol, M. Herzog and E. Jabara, Finite groups whose noncentral commuting elements have centralizers of equal size, Bull. Aust. Math Soc, 82 (2010), [15] I. M. Isaacs, Finite group theory, rad. Stud. Math, vol. 92, Amer. Math. Soc, Providence, RI, [16] The AP roup, AP-roups, Algoritms, and Programming, version , (2007),( [17] S. M. Jafarian Amiri and H. Madadi, On the maximum number of the pairwise noncommuting elements in a nite group, J. Algebra Appl, (2016), Vol. 16, No. 1 (2017) (9 pages).

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